density of (rl,r2,...,rD) given i, i.e. decide i such that
2 (1)
7- 1
is minimum. That sum is also equal to Writ) sl(f)||2, where II II is derived from the
innerproduct defined above, so that the demodulator essentially performs minimum distance
The resulting probability of error can be shown to depend only on the distances \\sl sJ\\2
between signals at the channel output, i.e. on
f\S'(f) -
SHf)\2 \H(f)\2/K(f)df
leading to the obvious conclusion that the signals sl (t) should differ at frequencies where H(f) is
large (i.e. in the channel passband).
K(f) is often assumed to be constant in that band, so that the innerproduct x,y is simply
the usual innerproduct
fx(t)y(t)dt .
The previous channel model is not accurate. Non linear and time varying distortions occur and
noise is not always Gaussian. Nonetheless modems typically exhibit the structure just derived:
first "filtering" elements and samplers compute the projection of the received waveform. They are
followed in turn by a non linear decision device that chooses the signal closest to the received
The signal space S has usually 1 or 2 dimensions, the basis functions being of the form x(t)
cos(27rfct) and x(t) sm(2irfct), where x(t) has a Fourier transform confined to the low
frequencies. The usual telephone lines have a passband between 300 and 3000 Hz, so that fc is
typically 1650 or 1800 Hz. n is 2, 3, 4, or 6 depending on whether data is transmitted at 2400,
4800, 9600 or 14400 bits per second.
3. TRANSMISSION OF SEQUENCES. The number N of binary digits that must be
transmitted by the modem is usually very large so that there is an enormous (2^) number of
waveforms, and the theory developed in the previous section cannot be implemented without more
structure being added.
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